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Theorem ralima 5338
Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Hypothesis
Ref Expression
rexima.x (x = (𝐹y) → (φψ))
Assertion
Ref Expression
ralima ((𝐹 Fn A BA) → (x (𝐹B)φy B ψ))
Distinct variable groups:   φ,y   ψ,x   x,𝐹,y   x,B,y   x,A,y
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem ralima
StepHypRef Expression
1 ssel2 2934 . . . 4 ((BA y B) → y A)
2 funfvex 5135 . . . . 5 ((Fun 𝐹 y dom 𝐹) → (𝐹y) V)
32funfni 4942 . . . 4 ((𝐹 Fn A y A) → (𝐹y) V)
41, 3sylan2 270 . . 3 ((𝐹 Fn A (BA y B)) → (𝐹y) V)
54anassrs 380 . 2 (((𝐹 Fn A BA) y B) → (𝐹y) V)
6 fvelimab 5172 . . 3 ((𝐹 Fn A BA) → (x (𝐹B) ↔ y B (𝐹y) = x))
7 eqcom 2039 . . . 4 ((𝐹y) = xx = (𝐹y))
87rexbii 2325 . . 3 (y B (𝐹y) = xy B x = (𝐹y))
96, 8syl6bb 185 . 2 ((𝐹 Fn A BA) → (x (𝐹B) ↔ y B x = (𝐹y)))
10 rexima.x . . 3 (x = (𝐹y) → (φψ))
1110adantl 262 . 2 (((𝐹 Fn A BA) x = (𝐹y)) → (φψ))
125, 9, 11ralxfr2d 4162 1 ((𝐹 Fn A BA) → (x (𝐹B)φy B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  wss 2911  cima 4291   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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